Puzzler 1 Imagine that you have three boxes, one containing two black marbles, one containing two white marbles, and the third, one black marble and one white marble. The boxes were labeled for their contents – BB, WW and BW – but someone has switched the labels so that every box is now incorrectly labeled . You are allowed to take one marble at a time out of any box, without looking inside, and by this process of sampling you are to determine the contents of all three boxes. What is the smallest number of drawings needed to do this?
Transcript
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Puzzler
Imagine that you have three boxes, one containing two black marbles, one containing two white marbles, and the third, one black marble and one white marble. The boxes were labeled for their contents – BB, WW and BW – but someone has switched the labels so that every box is now incorrectly labeled. You are allowed to take one marble at a time out of any box, without looking inside, and by this process of sampling you are to determine the contents of all three boxes. What is the smallest number of drawings needed to do this?
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Solution: One !!!
The key is that every box is incorrectly labeled.
Draw a marble from the box labeled BW
Assume it is black. – you now know which box has 2 black marbles.
Therefore you know the contents of the box labeled WW must be BW (it can’t be labeled correctly!)
Then the third box labeled BB must hold WW marbles
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Introduction to Audio and Music Engineering
Lecture 7
• Sound waves
• Sound localization
• Sound pressure level
• Range of human hearing
• Sound intensity and power
Period:
Frequency:
Angular Frequency: :Spatial Wavelength
Spatial:Wavenumber
Waves in Space and Time
TSeconds
Hertz (cycles per second)Radians per secondMetersRadians per meter
On a string the frequency of oscillation and the wavelength are connected through the speed of propagation of a bending wave.
c
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Sound wavesSound is a Longitudinal Wave: Disturbance varies along the direction of propagation.
Transverse wave: (string) Disturbance varies in a direction perpendicular to the direction of propagation.
pres
sure
c = 343 m/sec
Density of air = 1.21 kg/m3
f ≈ 20 Hz 20 kHz
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Question
What is the wavelength of a sound wave of frequency 20 Hz?
c = 343 m/sec
17.15 meters
c = 343 m/sec = 1125 ft/sec about 1 foot per millisecond
Remember this!
Wavelength = 1.715 cm @ 20 kHz
20 Hz ≤ f ≤ 20 kHz
17 m ≤ ≤ 1.7 cm
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Human ability to localize sound
Distance between human ears is ≈ 22 – 24 cm
f ≈ 1430 Hz
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Sound localizationf < 1500 Hz f > 1500 Hz
Wavelength is larger than distance between ears
Wavelength is smaller than distance between ears
Interaural Time Difference (ITD)
Interaural Intensity Difference (IID)
f < 1500 Hz f > 1500 Hz
Humans determine directionality of sound by two basic methods:
But there is some overlap of methods in the range 800:1600 Hz
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IID and ITD
IID: f ≤ 1500 Hz
Head shadows the sound at more distant ear.
ITD: f ≥ 1500 Hz
ITD time delay:22 cm 650 µsec
The shape of the outer ear (pinna) plays a significant role in 3D audio;Head Related Transfer Function: HRTF
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Sound Pressure Level
P is the measured pressurePref = 20 µPa (micro-Pascals)
1 Pascal = 1 Newton/meter2
1 Atmosphere = 14.7 lbs/in2 = 1.01 x 105 Pascals
Sound pressure of 20 µPa 0 dB SPL
Sound pressure of 20 Pa 120 dB SPL
1 Atmosphere = 194 dB SPL
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Range of Human Hearing
HOG ≈ 143 dB
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Adaptation of Human Hearing
At any given time we hear over a dynamic range of about 90 dB. Our auditory system adapts our hearing sensitivity to the average SPL – much like our eye adjusts to different lighting conditions.
0 dB SPL 140 dB SPL
90 dB
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Sound Intensity and Power
Sound Intensity: When a pressure wave propagates through air the air moves slightly.
